unit element

p.188 a partial order is asymmetric and transitive, but in
J.M.Howie, Fundamentals of Semigroup Theory, p.13 and in M.Petrich, Introduction to Semigroups, a (partial)order is asymmetric, transitive and reflexive,
p.22, a left identity is called a unit, but in J.B.Fraleigh, A Sirst Course in Abstract Algebra, p.210 a unit is an element which has a multiplicative inverse
p.192 an element which is greater than or equal to every element in the set is called a unit or an identity
why the ambiguity?